Numerical Solution I

Transcription

1 Numerical Solution I Stationary Flow R. Kornhuber (FU Berlin) Summerschool Modelling of mass and energy transport in porous media with practical applications October 8-12, 2018

2 Schedule Classical Solutions (Finite Differences) The Conservation Principle (Finite Volumes) Principle of Minimal Energy (Finite Elements) Adaptive Finite Elements (Fast) Solvers for Linear Systems Random Partial Differential Equations

3 Saturated Groundwater Flow (Darcy Equation) p: pressure S 0 p t = div(k p)+f S 0 = ρg n p 0: specific storage coefficient K = (K 1,K 2,K 3 ) : Ω R 3,3 : hydraulic permeability f = ρgdivk 3 +gf: gravity and source terms

4 Darcy s equation: Darcy s and Poisson s Equation S 0 = 0: pressure-stable granular structure Poisson s equation: div(k p) = f K R 3,3 constant: K p(x) = p(kx) transformation of variables: p(x) p(kx) =: u(x) u = f Laplace operator: u = 3 i=1 u x i x i = u x1 x 1 +u x2 x 2 +u x3 x 3

5 Darcy s and Poisson s Equation Darcy s equation: S 0 = 0: pressure-stable granular structure div(k p) = f Poisson s equation: homogeneous soil: K R 3,3 K p(x) = p(kx) transformation of variables: p(x) p(kx) =: u(x) u = f Laplace operator: u = 3 i=1 u x i x i = u x1 x 1 +u x2 x 2 +u x3 x 3

6 Classical Solution of Poisson s Equation Poisson s equation: Ω R d (bounded domain) u = f on Ω + boundary conditions on Ω boundary conditions (BC): u = g pressure BC (Dirichlet BC, 1. kind) αu+β nu = g transmission BC (Robin BC, 3. kind) Theorem (well-posedness) Assumption: Ω, f, g sufficiently smooth. Assertion: There exists a unique solution u C 2 (Ω) C(Ω) and u depends continuously on f, g.

7 flux boundary conditions: Ill-Posed Problems n u = g outflow BC (Neumann BC, 2. kind) no uniqueness: u solution = u+c solution for all c R necessary for existence: Ω f dx+ Ω g dσ = 0 (Green s formula) different BCs on disjoint subsets Γ D Γ N Γ R = Ω allowed Caution: Γ D too small, e.g. Γ D = {x 0,x 1,..,x m }, no uniqueness!!

8 flux boundary conditions: Ill-Posed Problems n u = g outflow BC (Neumann BC, 2. kind) no uniqueness: u solution = u+c solution for all c R necessary for existence: Ω f dx+ Ω g dσ = 0 (Green s formula) different BCs on disjoint subsets Γ D Γ N Γ R = Ω allowed Caution: Γ D too small, e.g. Γ D = {x 0,x 1,..,x m }, no uniqueness!!

9 flux boundary conditions: Ill-Posed Problems n u = g outflow BC (Neumann BC, 2. kind) no uniqueness: u solution = u+c solution for all c R necessary for existence: Ω f dx+ Ω g dσ = 0 (Green s formula) different BCs on disjoint subsets Γ D Γ N Γ R = Ω allowed Caution: Γ D too small, e.g. Γ D = {x 0,x 1,..,x m }, no uniqueness!!

10 rectangular mesh with mesh size h: Finite Differences Ω h Ω h o

11 Finite Difference Approximations x N 1. order forward and backward: x W x Z x O D + 1 U(x Z) = U(x O) U(x Z ) x O x Z x S D 1 U(x Z) = U(x Z) U(x W ) x Z x W δω 2. order central finite differences: ( 2 D 11 U(x Z ) = x O x W D + 1 U(x Z ) D1 U(x Z) )

12 Finite Difference Discretization (Shortley/Weller) discrete Laplacian: h U(x) = D 11 U(x)+D 22 U(x) discrete problem: h U = f on Ω h, U(x) = g on Ω h linear system: A FD U = b Theorem (Convergence): Assumption: u C 3 (Ω). Assertion: max U(x) u(x) = O(h) x Ω h

13 piecewise constant permeabilities: Heterogeneous Media K = { k1 x Ω 1 k 2 x Ω 2 Ω = Ω 1 Ω 2 Γ no classical solution of: div(k u) = f on Ω finite difference discretizations lead to interface problems interface problems require finite difference approximations of flux across the interface

14 piecewise constant permeabilities: Heterogeneous Media K = { k1 x Ω 1 k 2 x Ω 2 Ω = Ω 1 Ω 2 Γ no classical solution of: div(k u) = f on Ω finite difference discretizations lead to interface problems interface problems require finite difference approximations of flux across the interface

15 Conservation Principle conservation of mass in Ω Ω: K(x) u(x) dσ + f(x) dx = 0 Ω n Ω regularity condition: K u C 1 (Ω) d Green s formula: K(x) Ω n Ω u(x) dσ = div(k(x) u(x)) 1 dx+ K(x) u(x) 1 dx Ω Darcy s equation: Ω div(k(x) u(x))+f(x) dx = 0 Ω Ω

16 Finite Volumes finite dimensional ansatz space: S h, dim S h = n, v Ω = 0 v S h finite decomposition of Ω into control volumes Ω i : Ω = n i=1 Ω i finite volume discretization: u h S h : K n u h dσ = f dx i = 1,...,n Ω i Ω i linear system: choice of basis: S h = span{ϕ i i = 1,...,n} Au h = b, a ij = K n ϕ j dσ, b i = f dx Ω i Ω i solution: u h = n i=1 u iϕ i, u h = (u i )

17 Choice of Ansatz Space triangulation: T h = {T T triangle}, Ω = T T h T p q linear finite elements: S h := {v C(Ω) v T linear T T h, v Ω = 0} nodal basis: λ p (q) = δ p,q, p,q N h

18 Vertex-Centered Finite Volumes control volumes: linear connection of barycenters Ω p p linear system: A FV u h = b FV, u h = p N h u p λ p

19 Principle of Minimal Energy quadratic energy functional: J(v) = 1 2 a(v,v) l(v), minimization problem: a(v,w) = Ω K v w dx, l(v) = Ω fv dx variational formulation: u H 1 0(Ω) : J(u) J(v) v H 1 0(Ω) u H 1 0(Ω) : J (u)(v) = a(u,v) l(v) = 0 v H 1 0(Ω) weak solution u H 1 0(Ω) (Sobolev space)

20 Weak Versus Classical Solution regularity condition: K u C 1 (Ω) d Green s formula: 0 = a(u,v) l(v) = Ω (div(k u)+f)v dx+ Ω ( K n u) v dσ suitable choice of test functions v C 1 0(Ω): div(k u) = f on Ω

21 Ritz-Galerkin Method finite dimensional ansatz space: S h H 1 0(Ω), dim S h = n Ritz-Galerkin method linear system: u h S h : J (u h )(v) = a(u h,v) l(v) = 0 v S h choice of basis: S h = span{ϕ i i = 1,...,n} Au h = b, a ij = a(ϕ j,ϕ i ), b i = l(ϕ i ) solution: u h = n i=1 u iϕ i, u h = (u i )

22 Finite Element Discretization triangulation: T h = {T T triangle}, Ω = T T h T p q linear finite elements: S h := {v C(Ω) v T linear T T h, v Ω = 0} nodal basis: λ p (q) = δ p,q, p,q N h linear system: A FE u h = b FE

23 Finite Elements Versus Finite Volumes Theorem (Hackbusch 89) A FV = A FE, b FV b FE 1 = O(h 2 ) Corollary: u FV h u FE h 1 = O(h 2 ), v 2 1 = Ω v 2 + v 2 dx in general: finite volumes: finite element discretization with suitable test functions finite elements: finite volume discretization with suitable numerical flux

24 Stability maximum principle: u 1 = f 1, u 2 = f 2 on Ω, u 1 = u 2 = 0 on Ω no discrete maximum principle: f 1 f 2 = u 1 u 2 A FE u 1 = b FE 1, A FE u 2 = b FE 2 b FE 1 b FE 2 u 1 u 2 bad angles of triangles might cause oscillations! remedy: Delaunay triangulation T h

25 Stability maximum principle: u 1 = f 1, u 2 = f 2 on Ω, u 1 = u 2 = 0 on Ω no discrete maximum principle: f 1 f 2 = u 1 u 2 A FE u 1 = b FE 1, A FE u 2 = b FE 2 b FE 1 b FE 2 u 1 u 2 bad angles of triangles might cause oscillations! remedy: Delaunay triangulation T h

26 Stability maximum principle: u 1 = f 1, u 2 = f 2 on Ω, u 1 = u 2 = 0 on Ω no discrete maximum principle: f 1 f 2 = u 1 u 2 A FE u 1 = b FE 1, A FE u 2 = b FE 2 b FE 1 b FE 2 u 1 u 2 bad angles of triangles might cause oscillations! remedy: Delaunay triangulation T h

27 Stability maximum principle: u 1 = f 1, u 2 = f 2 on Ω, u 1 = u 2 = 0 on Ω no discrete maximum principle: f 1 f 2 = u 1 u 2 A FE u 1 = b FE 1, A FE u 2 = b FE 2 b FE 1 b FE 2 = u 1 u 2 bad angles of triangles might cause oscillations! remedy: Delaunay triangulation T h

28 Discretization Error Galerkin orthogonality: a(u u h,v) = 0 v S optimal error estimate: u u h = inf v S h u v, v 2 = a(v,v) (energy norm) ellipticity: α v 1 v β v 1, α,β > 0, v H 1 0(Ω) quasioptimal error estimate: u u h 1 β α inf v S h u v 1 estimate of the discretization error inf u v 1 c u 2 h = O(h), c = c(t h ) v S h

29 Adaptive Multilevel Methods Mesh T := T 0 Local refinement T := Ref (T ) no Discretization w.r.t. T Multigrid solution Error < TOL yes coarse grid generator finite element discretisation (iterative) algebraic solver a posteriori error estimate local refinement indicators local marking and refinement strategy quasioptimal error estimate: u u j 1 cn 1/d j

30 Adaptive Multilevel Methods Mesh T := T 0 Local refinement T := Ref (T ) no Discretization w.r.t. T Multigrid solution Error < TOL yes coarse grid generator finite element discretisation (iterative) algebraic solver a posteriori error estimate local refinement indicators local marking and refinement strategy quasioptimal error estimate: u u j 1 cn 1/d j

31 Phase Transition

32 Hierarchical A Posteriori Error Estimators extended ansatz space: Q h = S V h (hopefully) better discretization: u Q h Q h : a(u Q h,v) = l(v) v Q h basic idea: u Q h u h 1 u u h 1 algorithmic realization: basis of V h : V h = span{µ e e E h } (quadratic bubbles) weighted residuals: η e = r(µ e) a(µ e,µ e ), r(µ e) = l(µ e ) a(µ e,µ e ), e E h Theorem (Deuflhard, Leinen & Yserentant 88) The saturation assumption: u u Q h 1 q u u h 1, q < 1, implies the error bounds: cη u u h 1 Cη, η = e E h η e

33 Adaptive Refinement Strategy local errors have local origin (wrong for transport equations!): η e θ = mark all T with T e for red refinement! e e green closures:

34 Stable Refinement in 3D red refinement: What to do with the remaining octahedron? Learn from crystallography (Bey 91)!

35 Stable Refinement in 3D red refinement: What to do with the remaining octahedron? Learn from crystallography (Bey 91)!

36 Linear Algebraic Solvers stiffness matrix A = (a(λ p,λ q )) p,q Nh A is symmetric, positive definite and sparse condition number: 1 o(1) κ(a) = λ max(a) λ min (A) O(h 2 ) A is arbitrarily ill-conditioned for h 0 Corollary: Use iterative solvers for small h or, equivalently, large n (n )

37 Conjugate Gradient (CG) Iteration Underlying idea: inductive construction of an A-orthogonal basis of R n Algorithm (Hestenes und Stiefel, 1952) initialization: U 0 R n, r 0 = b AU 0, e 0 = r 0 iteration: U k+1 = U k +α k e k, α k = r k,r k e k,ae k update: r k r k+1, e k e k+1 error estimate: U U k 2ρ k U U 0, ρ = κ(a) 1 κ(a)+1 Corollary: Slow convergence for κ(a) 1 Vorkonditionierer: B leicht invertierbar, κ(b 1 A) κ(a)

38 Preconditioned Conjugate Gradient (CG) Iteration Underlying idea: inductive construction of an A-orthogonal basis of R n Algorithm (Hestenes und Stiefel 1952) initialization: U 0 R n, r 0 = B(b AU 0 ), e 0 = r 0 iteration: U k+1 = U k +α k e k, α k = r k,br k e k,ae k update: r k r k+1, e k e k+1 error estimate: U U k 2ρ k U U 0, ρ = κ(ba) 1 κ(ba)+1 Corollary: Fast convergence for κ(ba) 1 preconditioner B: optimal complexity O(n j ) and B A 1

39 Subspace Correction Methods basic idea: solve many small problems instead of one large problem subspace decomposition: S h = V 0 +V 1 + +V m Algorithm (sucessive subspace correction) Xu 92 w 1 = u k v l V l : a(v l,v) = l(v) a(w l 1,v) v V l w l = w l 1 +v l u k+1 = w m Gauß-Seidel iteration: V p = span{λ p } ρ h = 1 O(h 2 ) Jacobi-iteration: parallel subspace correction ρ h = 1 O(h 2 ) domain decomposition, multigrid,... ρ h ρ < 1 BPXW 93

40 Multigrid Methods hierarchy of grids (adaptive refinement!): T 0 T 1 T j = T h hierarchy of frequencies: S 0 S 1 S j = S h λ (k) p λ (j) p multilevel decomposition: S h = j k=0 p N k V p (k), V (k) p = span{λ (k) p } multigrid method with Gauß-Seidel smoothing and Galerkin restriction

41 Poisson s Equation on the Unit Square Ω = (0,1) (0,1), f 1, V(1,0) cycle, symmetric Gauß-Seidel smoother coarse grid T 0 approximate solution convergence rates: j=2 j=3 j=4 j=5 j=6 j=7 ρ j

42 Approximation History (L-Shaped Domain) levels nodes iterations est. error est. error/n 1/2 j

43 Uncertain Data uncertainty of permeability, source terms, boundary conditions,... by lack of sufficiently accurate measurements by external sources random Darcy equation: S 0 p t (t,x,θ) = div(k(x,θ) p(t,x,θ))+f(x,θ), θ Θ, probability space (Θ,A,P).

44 Random Partial Differential Equations random Darcy equation: S 0 p t (t,x,θ) = div(k(x,θ) p(t,x,θ))+f(x,θ) desired statistics of solutions: expectation value E[p] := Θ pdp variance E[(p E[p]) 2 ] probability of particular events P[ p(t,x,θ)dx < 0 for t < 1] Ω

45 Numerical Approximation of Random PDEs Monte-Carlo Method: (i) generate N independent, P-equidistributed samples θ 1,...,θ N. (ii) compute approximations of the N solutions for the samples θ 1,...,θ N. (iii) compute the corresponding expectation value, variance,... advantage: always feasible and simple disadvantage: convergence rate 1 N = N has to be very large multilevel Monte Carlo methods: solve stochastic problems with deterministic computational cost

46 Numerical Approximation of Random PDEs Monte-Carlo Method: (i) generate N independent, P-equidistributed samples θ 1,...,θ N. (ii) compute approximations of the N solutions for the samples θ 1,...,θ N. (iii) compute the corresponding expectation value, variance,... advantage: always feasible and simple disadvantage: convergence rate 1 N = N has to be very large Multilevel Monte Carlo methods: solve random PDEs with almost deterministic computational cost!